2019-04-22 – Between Midrash and Logic – Lesson 11

Table of Contents

• [0:01] X, Y, and the irrelevant parameters
• [3:00] The microscopic parameters in the table according to the Talmudic text in Bava Kamma
• [4:30] The green frog analogy and the connection to color
• [12:35] A pirkha— a situation of open possibilities
• [15:43] Choosing the simpler explanation in order to get a single result
• [23:01] Building a model with parameters and a mathematical proof
• [29:27] Establishing an order relation between columns B and A
• [31:10] Choosing a solution when B is stronger than A
• [34:28] Translating the diagram into the languages of tables
• [35:46] The pirkha in the kal va-chomer model
• [38:38] The prohibition on “turning around” a kal va-chomer
• [41:15] Summary and comments on the complexity of the parameters

Summary

General Overview

The text states that after X and Y were clarified as irrelevant, the remaining question is whether Z obligates even without the exemption for vessels and for a person, and from there the possibility arises of a conclusion opposite to the accepted one—that there is none of the exemptions at all. The text presents the Rosh as a third approach that sets up a “dominant teacher” in the common side, while the second teacher serves only to remove a pirkha, and everything depends on the specific situation rather than on a universal principle. The text develops a formal model of kal va-chomer and analogy through “microscopic parameters” hidden behind the table, and formulates an algorithm in which one chooses the correct filling according to the simpler explanation. The text defines a pirkha as a situation in which two explanations are equally simple, so the decision remains open, and through this explains why you cannot “turn around” a kal va-chomer.

X, Y, Z, and the exemption for vessels and for a person

The text says that after it became clear that X and Y are irrelevant, it remains to examine whether Z obligates even without the exemption for vessels and for a person, and it brings a case in which there is Z and yet there are still no exemptions for vessels and for a person. The text concludes that if there is liability in Z, there is no reason to grant there an exemption for vessels and for a person, and the exemption apparently stems from a particular feature that does not exist here. The text argues that from here one can reach a conclusion opposite to the accepted one: that in such a case there will be none of the exemptions at all.

The Rosh’s approach and the dominant teacher in the common side

The text presents the Rosh as a third approach holding that there is an exemption of “one of the teachers,” meaning the dominant teacher. The text states that according to the Rosh, the dominant teacher is the main one not because it comes first but because it is more similar, while the second teacher serves only to remove a possible pirkha. The text says that after removing the pirkha, one proves that X is irrelevant and goes back to learning from Z to Z by the regular analogy. The text adds that there are tools for deciding what is more similar through reasoning, and that there may be situations in which the two teachers are equally similar, in which case the Rosh would say something different, because this depends on the situation and not on a universal principle; in that way one can solve the later authorities’ questions on the Rosh.

The microscopic parameters behind the table in Bava Kamma and throughout the Talmud

The text says that in Bava Kamma the Talmudic text is an exceptional case in which they put on the table the microscopic parameters hidden behind the table, whereas in most passages throughout the Talmud they raise pirkhas and all kinds of arguments without spelling out the parameters. The text states that the table presents only monetary liabilities, while properties such as its intent to damage, the beginning of its making being for damage, and its way being to go and damage in the case of fire, are the microscopic parameters that generate the differences. The text argues that there always has to be something hidden behind the table in order to allow a kal va-chomer, like the “seasoning” that connects a preference for jazz to a preference for literary fiction, and that this is the insight from which “everything rolls onward.”

Searching for a model for kal va-chomer through alpha and beta

The text begins from a kal va-chomer as a data table with three pieces of data and one missing entry, and assumes that the correct filling should arise from the explanation behind the other three pieces of data. The text explains analogy through the example of “the frog is green… because they are both frogs,” in order to establish a relevance connection between a microscopic parameter and an observable property, and accordingly searches for a “model” that explains the table. The text first proposes a single parameter, alpha, and derives from it conclusions about the intensity of alpha required in the public domain and in the damaged party’s courtyard, and about the alpha values of tooth and foot and of horn, until it seems that one arrives at an answer of liability.

The failure of the one-parameter model, the move to a two-parameter model, and defining a pirkha as an open possibility

The text says that the one-parameter model is an “illusion,” because alpha equal to zero is the absence of an explanation, and the absence of alpha cannot be the factor that includes liability in the damaged party’s courtyard. The text adds a parameter beta and concludes that tooth and foot have beta, which allows liability in the damaged party’s courtyard even when alpha is zero, whereas horn may have alpha equal to one but beta equal to zero, and therefore the result could come out the opposite way. The text rejects the possibility of being satisfied with the fact that “there is a possibility” of liability, and states that kal va-chomer has to be a proof; and when one can reach two results equally, that is a situation of pirkha. The text defines a pirkha as a situation in which it has not been proven that the filling is one, because it could be either one or zero and both possibilities remain open—not as a situation of a counterproof that decisively proves zero.

One non-binary parameter: one and two

The text proposes a third possibility: a single parameter, alpha, that is not binary but takes the values one and two, and explains that the logic of kal va-chomer requires an axis on which one is stronger than the other, so “this one has two and that one has one.” The text infers that in order to obligate in the public domain one needs two, and therefore tooth and foot, which have one, do not obligate there but do obligate in the damaged party’s courtyard, where one is enough; and horn, which does obligate in the public domain, is thereby proven to have two, and therefore all the more so obligates in the damaged party’s courtyard. The text sums up that there are now three possible explanations of the table, two leading to a result of one and one leading to zero, and asks which explanation should be chosen.

The rule of simplicity: fewer parameters are better than adding another parameter

The text states that the choice is not “mindless” but needs justification, and sets up a rule of choosing the simplest explanation. The text formulates that an explanation with one parameter and two values is simpler than a two-parameter explanation, and adds that the justification of this rule rests on the empirical fact that kal va-chomer “works,” and therefore, translated into mathematical language, the complexity of adding a parameter complicates things more than increasing the number of values of the same parameter. The text says that here he is going back and forth between intuition and building the model, and presents this as constructing a formalization, similar to Aristotle, who built logic out of observing the way natural thinking works.

An algorithm for filling the table according to the simpler explanation

The text proposes an algorithm in which one assumes once that the missing entry is one and once that it is zero, and for each assumption searches for the simplest explanation. The text states that the filling whose explanation is simpler is the “winner” and reflects the correct filling. The text shows that in one table one can explain the filling of one through a single parameter with values one and two, whereas in the table with a filling of zero there is no way to explain it with a single parameter and one is forced into two parameters because of independence between the columns; therefore the filling of one is preferable.

The meaning of alpha in actions versus domains

The text emphasizes that the meaning of the value of alpha flips between “actions” and “domains.” The text says that in actions, a higher alpha value means a stronger action that succeeds in creating liability even where a weaker action would not. The text says that in domains, a higher alpha value means that it is harder to create liability there, because a higher value is needed in order to apply liability, and he compares this to someone with more money who can buy more, as opposed to a higher price, which makes buying harder.

A pirkha on kal va-chomer: equivalent explanations and preventing a decision

The text takes a pirkha “off the cuff,” in which tooth and foot obligate and horn does not obligate, and applies the same algorithm to two tables—one under the assumption of a filling of one and one under the assumption of a filling of zero. The text states that a pirkha is not supposed to provide a simpler explanation for zero, but rather to bring things to the point where the two explanations are equivalent, so that one cannot choose. The text concludes that in the case of the pirkha, both fillings lead to a two-parameter explanation at the same level of complexity, so openness remains—and that is exactly what a pirkha is.

An order relation between columns, independence, and diagrams of alpha/beta

The text proposes a way to build models by identifying an order relation between columns: if B is stronger than or equal to A in every row, one marks an order relation, and if each one is more stringent than the other in a different row, then there is no relation at all. The text states that when there is an order relation, there are two basic possibilities for explaining the increased stringency: “two alpha” or “alpha and beta,” and if there is no further constraint, one chooses “two alpha” as simpler. The text adds that in certain tables there is no choice and one must add another parameter, and explains that independence creates two axes that do not speak to one another, like two different seasonings such as love of jazz and love of literature.

Why you cannot turn around a kal va-chomer, and what pirkha does to relevance

The text argues that even if “you flip whatever you want,” a kal va-chomer is one argument, and once there is a pirkha, the two models are at the same level of simplicity, so it does no good to turn the formulation around through columns or rows. The text states that this explains why throughout the Talmud they do not turn around a kal va-chomer, because turning it around does not change the table, only the way one talks about it. The text interprets pirkha as meaning that the relation of stringency is correct but not relevant, because it lies in the parameter alpha, while there exists another feature, beta, in which the other side is more stringent, and the question is which parameter determines the liability.

A note on number of values versus number of parameters

The text acknowledges that sometimes the filling of zero has an advantage in that its two parameters are binary, while another explanation uses a single parameter with three values—zero, one, and two. The text concludes from this that at this stage, the number of values does not make an explanation less attractive, and indeed “does not change anything” compared to the number of parameters, and declares that this is a stage in the construction and that later he will “walk some of it back.”

So now why should I care about X and Y? X and Y turned out to be irrelevant. So now all that matters is just this, right? But Z obligates. You see that Z does not exempt vessels and a person. Here, there is Z, and there is no exemption for vessels and a person. So if here it has Z and he is liable, why should there be an exemption for vessels and a person? The exemption for vessels and a person apparently comes from this feature. That feature does not exist here. Right? So the conclusion actually comes out the opposite of what people think: really, none of the exemptions would apply. And the Rosh, by contrast, has a third approach. The Rosh claims that there is the exemption of one of the source cases, the dominant one. How did he understand the issue? He simply understood it the way I said before: that this is actually the real source case. Except that I have a refutation: what about this one, which has X? That proves that X is not relevant, right? After I proved that X is not relevant, I went back to learn from it. In the end, I’m learning from it. Is the dominant source case X? Because it is the main one, not because it is first. The common denominator? Is that the Rosh’s opinion? Yes. It is more similar. It is more similar in some straightforward sense. Would it have been less technical if we had started from the visual? It is not a logical connection; it is a connection of resemblance. It is more similar, therefore it is primary, and the second one serves me only in order to remove a possible refutation. Once I removed it, then X is irrelevant; that’s what I prove from here. After I remove it, I go back to learning from Z to Z. It’s just the ordinary analogy. Are there tools for deciding what is more similar? Yes. Reasoning. In this example it’s not—there isn’t—no, this is just an example. There the discussion is about derivations, not about the primary categories of damages. Here both are similar to the same degree. Right. Okay? So it could definitely also be different, by the way. There could be places where in a common-denominator derivation, both source cases are equally similar for the purpose of teaching, and there the Rosh would not say what he says. By the way, you can solve many difficulties raised by the later authorities (Acharonim) against the Rosh that way. Because it is not a general principle; it depends. It simply depends on the situation the Rosh is talking about, where it was clear to him that one of the source cases was more similar. It is the dominant one. And then the second one is only there to remove a refutation. But there may be other common-denominator cases where the two source cases are similar. Then he would say something else. It is not some universal principle. Okay?

Now why—why is this important? Because here in this case, and in Bava Kamma generally, you see that I’m playing with microscopic parameters. Right? These parameters will not appear in the table—notice that. X, Y, and Z will not appear in the table. In the table it will say: the ox is liable for payment, and here let’s say it is not liable for payment, whatever. And fire is liable for payment here, so certainly it will be liable for payment there. Fine? But the table only contains obligations of payment. Where do the features appear—its intent is to cause damage, its initial nature is to cause damage, its way is to go and cause damage in the case of fire, yes, or all kinds of things like that? Where do those features appear? Those features are the microscopic parameters hidden behind the table. The Talmud in Bava Kamma is an exceptional case because it puts the microscopic parameters on the table. It shows them to us. In most passages in the Talmud this does not happen. They make refutations, they do all sorts of moves, but they do not talk about what the microscopic parameters really are. I’ll show this. Okay? But this is a nice example because here you can see the form of thought. Will there always be hidden parameters? There always are, always, because if not, then it goes back to jazz and literature. You have to assume that the same ingredient causes the preference in love of jazz just as it causes preference in love of fine literature; otherwise there is no room, you cannot make an a fortiori inference, even though the table looks like this. The table looks like this. You still have to assume something hidden behind the table, that what causes these zeros and ones is the same factor as what causes this one. Okay? So there is something behind the table that is really generating the whole thing. And this is actually the most important insight. From here on, everything unfolds. Okay? Now let’s start unfolding it.

So let’s begin with a fortiori reasoning. Let’s begin with a fortiori reasoning. I am now looking for what to fill in here. I don’t know; I have three data points, and I ask myself what to fill in here. My assumption is that what goes here will be what follows from the explanation of the other three data points. If you remember, when we did analogy, I gave an analogy: the frog is green, this one is also a frog, so this one is also green. Why do I compare the two and say they are both green? Because they are both frogs. Meaning, the feature of being a frog is relevant to the green color. If it were not relevant, the analogy really would not be plausible. Right? The plausibility of the analogy is based on our assuming in advance that your being a frog is relevant to your having the color green. There is a relation of relevance between the microscopic parameter—you are a frog—and the feature we see with our eyes—that you are green. Fine? I am saying the same thing here. Let’s try to look for an explanation—later I’ll call it a model, okay? A model that explains this table. I want to look for a model.

So let’s see. Suppose there is a microscopic parameter—we’ll call it alpha, okay? And this alpha—let’s see—what alpha values do A and B have? So let’s say A is the weaker one, right? B is the stronger one. Right? We see that from here, the a fortiori structure of the columns. B is the stronger one. So A has alpha zero, and B—its alpha value is one. Fine? It has more of the microscopic parameter alpha, of that ingredient that causes liability for payment. So if this is not liable for payment here, and this is liable for payment here, that means that whatever causes liability for payment—which I do not yet know what it is—is present more here than here. It is more, in the language of the Mishnah, your property and under your responsibility to guard, than A is. Fine? In some sense.

Okay, now let’s continue unfolding it. So if A is zero, and it still manages to produce liability in B—let’s say, say, this is tooth and foot, this is horn, this is the public domain, and this is the injured party’s courtyard. Okay? So if tooth and foot manages to create liability in the injured party’s courtyard, what does that mean? That in order to create liability in the injured party’s courtyard, you do not need alpha. So about the injured party’s courtyard too I say its alpha is zero. But notice—in a different sense from the zero here. The zero here is: what power do you have? The zero here is: what power is required in order to apply the law to you. In a moment I’ll sharpen that more.

Look, what happens in the public domain? Here, of course, you need alpha value one, and therefore if you have zero, it is not enough. You do not have enough alpha-strength to impose liability here. Right? I’m not yet asking what alpha is in all this; I’m just assuming there is such a thing. And in the injured party’s courtyard you do have enough. A fact—you succeed in obligating there. That means that for the injured party’s courtyard, alpha equal to zero is enough to obligate. But for the public domain, I’m missing something. Apparently alpha equals one, in some arbitrary unit; not important right now. Okay?

Now I say: let’s move to horn. Horn does manage to impose liability in the public domain, right? If it manages to impose liability in the public domain, then clearly it has alpha equal to one. That already follows. I begin—this is the assumption—and from here on everything unfolds. Just this. Now look: regarding the public domain, the fact that it is one is a result. Regarding the injured party’s courtyard, the fact that it is zero is also a result. Here, the fact that it is one is also a result, right? Because if B manages to obligate in the public domain, and the public domain requires one in order to obligate, then if B manages to do the job, that means B has alpha value one. Alpha-strength of one, right? Therefore it succeeds in applying liability. And now I ask myself: tell me, will horn be liable in the injured party’s courtyard or not? Obviously yes. A fortiori. Why a fortiori? Because in the injured party’s courtyard, even zero is enough in order to obligate, and that one has even one—more than enough. So all the more so it will be liable. Therefore the answer is one. Fine?

But there is a certain problem here. What is the problem? There could be a beta. But I am assuming the simplest possible model. I’ll define this more precisely later. If I have a model with one parameter, certainly I am not going to go to two. The problem is that I do not really have a one-parameter model. What is written here is an illusion. Why? Because tooth and foot do not have alpha. Their alpha value is zero. So good grief, then what generates the liability in the injured party’s courtyard? It isn’t alpha, because it does not have alpha. So what generates the liability in the injured party’s courtyard? What, the absence of alpha generates liability? The absence of alpha also means the absence of millions of other parameters. Absence does not generate liability. There has to be something because of which the liability is created, right? Wait, wait, wait. In a moment I’ll get to that. You’re right, but in a moment. I want to proceed simply, didactically. Okay?

So I say, of necessity, this is simply an illusion. There is another parameter, beta. And the parameter beta is present in tooth and foot and absent in horn. Okay? And now let’s correct it. Tooth and foot have alpha and no beta; horn has alpha and no beta. Okay? And now let’s start unfolding. In the public domain, tooth and foot do not manage to impose it. They do not manage to impose liability there. Why? Because what is needed is alpha equal to one. Right? Here there is alpha equal to one, and if this succeeds in imposing it, then beta equal to zero is enough. Because otherwise this too would not have succeeded in obligating here. Right? So beta equal to zero is enough. Okay. What happens here in the injured party’s courtyard? In the injured party’s courtyard, tooth and foot do manage to impose liability. That means that beta equal to one is enough even with alpha equal to zero. Okay? Horn has alpha equal to one but beta equal to zero, but here beta equal to one is needed in order to impose liability. So the result is zero. So it is not true that the conclusion is one. The result is zero. Why is the result zero? Because when I do the calculation, alpha equal to zero is not an explanation. Alpha equal to zero is the absence of an explanation. Meaning that in order to impose liability in the injured party’s courtyard, it is not enough for me to write that alpha equals zero. There has to be some other parameter that is the cause of liability in the injured party’s courtyard. Let’s call it beta. And its value must be one. But if its value has to be one and tooth and foot manage to create liability, then tooth and foot have beta value one. Otherwise they would not have succeeded in obligating.

Wait, wait. The alpha value of horn is one, right? I see that from here, because it manages to impose liability in the public domain. Yes. Okay. The beta value of horn—I have no idea. You cannot know anything from here. Right? You could seemingly say that alpha overrides beta—meaning just make another a fortiori step between alpha and beta, and so on to infinity. No. You can also arrive at one if you assume beta equals one here, which you do not have to assume, and you can arrive at the result zero. If you can arrive at both results equally well, then you have no proof. What I am saying here is different: you could say that just as we assumed—what we did with zero and one, just as we made the first a fortiori move between A and B above, you can do the same between alpha and beta. If alpha says: not in the public domain but yes in the injured party’s courtyard, then with beta too, all the more so, if beta says yes in the public domain then it is stronger than alpha. Who says? And so on. Who says? What do you mean, and so on? It could be and it could be not. It could be and it could be not. But a fortiori reasoning is supposed to be a proof, not to suggest a possibility. The fact that you suggest to me that there is a possibility that you would be liable here is not enough. The a fortiori argument has to show me that you must assume there is a one here. Okay? So if there is a possibility of one and a possibility of zero, that is the state of a refutation, as I will define in a moment. A refutation means the question remains open. The entry could be one, it could be zero; both options remain open. That is called a refutation. A refutation is not a case where I proved that there is a zero here. Because if I proved there is a zero here, that is not a refutation; that is a counter-proof. A refutation means: you did not prove that there is a one here. Why? Because it could be either one or zero; both options are open. That is called a refutation. Therefore a situation like this is a refutation, not an a fortiori argument. It is impossible to prove that the entry there is one.

So what happens? We’re in trouble. But it isn’t true that we’re in trouble, don’t worry. I am now going to suggest a third possibility to explain this. Maybe without erasing. The third possibility says: there is only one parameter, but its values are one and two, not zero and one. So we move to a non-binary situation? Yes. Two values that are not zero, of the parameter alpha. And understand: the logic of a fortiori reasoning requires this. By definition, the logic of a fortiori reasoning says there is some axis along which one is stronger than the other. Meaning that this one has two and that one has one. And now let’s start unfolding. Then it is not a situation of either there is or there isn’t? Exactly. It is a situation of there is and there is—how much there is. Exactly. Okay.

Now let’s start unfolding again. Let’s start over. Then we say as follows: if the alpha of tooth and foot is one, what is needed in order to impose liability in the public domain? What is the question? What is the question? I am saying: I now assume this picture. Fine? Now I begin drawing conclusions here. Two are needed for the public domain. Not that I need a decisive proof—there must be two here. Because if it were one, that is obvious, because if it were one then there would also have to be a one here. We have enough alpha if one were enough. Clearly, what is needed here is alpha with value two, and tooth and foot do not have enough alpha for that, therefore they do not succeed in obligating. Right? What happens here? Here they do succeed in obligating. What does that mean? That one is enough. Right? Now let’s move over here. In the public domain, horn succeeds in obligating. What does that mean? Again, I am assuming only this, I am not assuming two. I am assuming only this, and then I have proved that horn has two. Right? Because the fact is that it succeeds in imposing liability in the public domain, and in the public domain two are needed in order to impose liability. So that means it has two. Now let us return and ask the question: does horn succeed in obligating in the injured party’s courtyard? All the more so, certainly yes. Because for the injured party’s courtyard, one is enough in order to obligate, and horn has stronger force, so certainly it succeeds in obligating.

So in fact, up to this point I have shown three possible explanations of the table, right? I presented three explanations, all three of which explain the table, and one of them leads to the opposite result. Two of them lead to result one, and one of them leads to result zero. Okay, which explanation do I choose? Obviously I will want to choose this explanation so that the result will be one. But that is no trick—you have to justify why I choose this explanation, because otherwise who says this is the correct explanation? Maybe the explanation that leads to zero? That is exactly what “what about this one, which has…” means; it is exactly like rotating the matrix. If every explanation has a different result, then it is not explanatory. No, not true, not true. The claim is that I am really choosing, as in everything else, the simplest explanation. The two previous explanations assumed a two-parameter model, alpha and beta, right? I am offering you a simpler explanation. With a single parameter I explain the whole table. So which is the correct explanation? The simpler one, right? So now when I take the simplest explanation, this is the simplest explanation, and if so the result must be one. But it is not necessarily simpler, because although you are using fewer parameters, you are now assuming that increasing the valency does not complicate the model as much as adding a parameter. That is what I am assuming here, right? I am now going backward. I am really saying: okay, the empirical datum is that a fortiori reasoning works. It is reasonable, right? So it is clear that when translated into the mathematical language of this model, apparently increasing the number of values of a parameter complicates the theory less than adding another parameter. Meaning: an explanation with two parameters is more complicated than an explanation with two values of one parameter. That is a conclusion. I did not assume it; I am saying: if a fortiori reasoning works, then apparently the translation into parameters is subject to this rule—that the complexity of adding a parameter is greater, or more complicated, than the complexity of increasing the number of values of a single parameter, and therefore this answer is the preferable one.

Now I will do this in a more general way, and I say this: we now have a table like this. Now I am proposing an algorithm, okay? The algorithm says this: I begin to analyze a table like this, and afterward I analyze another table like this. I propose an explanation—notice, I am no longer proceeding the way I did before; you will see later that this is simpler. It is the same thing, but as a method this is simpler and clearer. I assume two possibilities, either one or zero. Let’s see which of them gives me a simpler explanation. This is the reverse way of looking at what I did before, okay? Whichever one gives a simpler explanation probably reflects the correct entry. I assume entry one; here I assume entry zero. I will look for an explanation under the assumption that the entry here is one; I will look for an explanation under the assumption that the entry here is zero; and let’s see which explanation is simpler. The simpler explanation wins. Okay, so let’s see. So an explanation that gets there faster? No, an explanation that is simpler—that is, with fewer parameters or with lower valency. But that was the model a second ago here, and how do I measure simpler? So now I am going in the opposite direction, I said. I am looking for the simplest explanation, okay? But I know that the correct entry here is one. So that will give me a hint that I should assume that the model obtained for the upper table is in some sense simpler. Now this is a hypothesis. I will test whether it works later, and that is how little by little I build the model. Okay? This is a model that is not built a priori. If I could build it a priori, everything would be very easy. I build it in comparison with my intuition. By the way, every logical formalization is built like this. How did Aristotle build the formalization of ordinary logic? He looked at how we think naturally. And from that he formalized logic, right? I am doing the same thing. I am looking at how we think in non-deductive thinking in this case—whether a fortiori reasoning or analogy or something like that. And I am formalizing how we think. Meaning: how we think is, for me, the datum. I am not going to argue with that. That is the correct datum. What I am trying to do is mechanize it. That is, to find a formal language that succeeds in performing what our thinking does intuitively. Okay?

So now I say this: I am looking for an explanation of this thing. So what is the explanation? Here I succeed in finding an explanation with one parameter. Right? Here one and here two. Right? That explains the table, agreed? If A has value one and here value two is needed, then it does not succeed here. If here value one is enough and it has one, then it succeeds here. Right? B has value two of the parameter, so for A, which requires value two, that is perfectly fine; and for B, where value one is enough, then if you have two that is certainly fine. Right? So that is the explanation of the table with entry one. This is an explanation with one parameter that can take two values—either one or two.

Okay? That is the explanation for the upper table. What happens in the lower table? In the lower table, no matter what you do, stand on your head, you do not have an explanation with a single parameter. You will not find one. No matter how many values you give it, you will not find one. You are forced to move to two parameters, and the reason is very simple—there is a straightforward independence between these two columns, right? This column is more severe than that one according to this parameter, but less severe than that one according to that parameter. Right? So that means there are two parameters, such that from the standpoint of one, this is better, and from the standpoint of the other, that is better. Right? That is the intuitive way to see it. In mathematics they say that these two rows are independent. Okay? You all look a bit glazed over. Yes, yes. Is this an explanation for what you say? The whole Mishnah in Shevi’it chapter 4, Mishnah 8, Beit Shammai and the Sages? Yes. You see there that even though ostensibly the first table is more correct, it did not work. It did not work, at least not in… No, no, there are opinions there. There it is half and one inside the table; I already said that last time too. That changes the picture completely. Know that if it is half and yes, then we will get to that. If it is one and zero, then it is always, always… I have not assumed anything here—what did I assume here? I drew the table; that is the solution. I did not assume anything about what A is, what B is. I assumed nothing. Every a fortiori argument in the Talmud looks like this, after all. Right? So clearly this is universal. But if it is refuted… Fine, soon we will see what happens when it is refuted. Don’t worry. Right now I am speaking about a fortiori reasoning; there are no refutations yet. Okay? I am going step by step. After that I will show what a refutation does; after that I will show what a common denominator does; after that we will examine a paradigm derivation; we will examine a refutation against a paradigm derivation. We will start climbing slowly. Right now I am beginning to build the model.

Okay? So now, in this model, try—you can try it on paper—you can prove it mathematically in a very simple way, but you can try it on paper: you will not succeed in finding here a model with one parameter that explains this table. There is none. You will be forced to use two parameters. Then in effect there is here a two-parameter model, alpha and beta. A, let us say, has alpha and no beta, and B has no alpha and has beta. And now correspondingly, A succeeds in imposing B. Right? So that means that B requires what A has. Okay? By contrast, it does not succeed in imposing A. Why? Because apparently B does succeed in impose that one. So that means it is zero-one, and therefore A also does not succeed in imposing it because it does not have enough beta; its beta is zero. Okay? So that is the explanation for the lower table. Meaning, this one has one-zero and that one has zero-one. This one requires zero-one and that one requires one-zero. What does that mean? It means that the model explaining entry zero is a more complicated model than the model explaining entry one. So that means entry one is the correct one. It is the same thing I explained before, only here it is already completely systematic. I no longer need all those attempts I made before, checking what comes out in the empty square. I simply fill in the square and then look for an explanation. The simpler explanation, taking into account the filling of the square as well, is the explanation I choose. Once that is the simpler explanation, for me the inference means that the correct entry here is one. Because it has a simpler explanation. Okay? So that is more or less the algorithm.

Now just a few remarks. First remark—the meaning, what I spoke about before, the meaning of the parameters. Look, when I say that tooth and foot have alpha with strength one, and horn has alpha with strength two—what does that mean? That horn is stronger than tooth and foot. Yes, even in a place where tooth and foot would not be liable, horn is stronger. It is possible to impose liabilities on it even in a place where this one cannot produce liabilities. Right? So in terms of the actions and the domains—regarding the actions, the higher the alpha, the stronger the action, in the sense that it is more capable of obligating. Okay? By contrast, here in the domains, the higher the alpha, the harder it is to obligate. Right? Because you need a higher alpha value in order to obligate. Do you see? Meaning, the situation here is the reverse, and you need to keep your head straight about this. In the actions, a higher alpha value means we are speaking about a stronger action, which most likely—exactly, which is more likely to succeed in obligating than an action with a lower value. By contrast, in the results or the domains, the bigger alpha is, the less you can obligate there, because you need a bigger alpha value. Just as someone who has more money is more likely to manage to buy more things, and the higher the price, the more likely it is that fewer people will manage to buy it. Right? Exactly the same thing. Okay? So that just has to be kept in mind.

Okay? Now let’s move on. The next stage is to check what happens when there is a refutation. Let us take this refutation as an example. Fine? This is on the side. Okay? On the side, tooth and foot obligate and horn does not obligate. Fine? Now I want to know what this addition of the refutation does to my a fortiori argument—how it breaks the whole thing. I return to exactly the same algorithm and I do the following. I take the first table—here I fill in a one. Okay, this is under the assumption that the entry is one, right? And this—now I’ll make another table here on the side—this is with entry zero. You see? Here there is zero and here there is one. What do I do now? I look for explanations for both tables, and I want to check which explanation is simpler in order to know which is the correct entry. What do I expect? That with a refutation there will be… no… less? That there will be two parameters instead of three? Meaning that the zero will have a simpler explanation. No, that the zero will have a more complicated explanation—three parameters. Worse. So what, after a refutation the entry is also one? But if there is a refutation, then I say the entry is not one. Maybe both will be the same? Exactly. I am not looking for the explanation of this one to be simpler—if so, that is not a refutation, that is a counter-proof. Right? A refutation is not a counter-proof. A refutation does not say that there is a zero here. A refutation says that the question remains open—you have not succeeded in proving. It is either zero or one; these are two equivalent explanations. Meaning, notice, I am constantly going with simple logic and with the model and comparing them. That is how I build the model. So now I want to look for explanations and see that these two explanations are equivalent. Therefore I cannot choose either one. So let’s look.

So now I have—maybe I’ll do it like this, look. Before I do it, I’ll just show you already the way to build these models without getting tangled up. Suppose there is here an a fortiori table. Let’s go back for a moment. Okay? An a fortiori table—what happens here? Notice. What happens here is A, B. Notice that B is always stronger than A in all the rows, right? Either stronger or equal. Okay? So let’s call that an order relation between the two columns. B is greater than or equal to A. Okay? I mark it like this. Okay? That is my notation. That is how I mark it. Fine? By contrast, where here there is zero… what happens? Different or equal? No. There is no relation between them at all. There is no relation between them at all, right? Neither one is greater than or equal to the other in all the rows. So for that I say there is no order relation between the columns. Okay? This will help us in more complicated tables, so it is worth noticing it here. For a fortiori reasoning you can manage even without it, but in more complicated tables it will already be much harder for you. Okay? So we said there is no relation between the two columns.

Now notice: once B is stronger, we already saw what comes out, right? What happens when B is stronger? It means that if this is alpha, then this is two alpha. Right? There was a question: is there no meaning at all to the fact that there is an inversion there between A and B? No, there is no reverse relation. There is a relation in one direction, and in the second row a relation in the other direction. So there is no reverse relation either. You cannot write A greater than or equal to B; that is also not true. There is no relation between them at all. In the place where A is zero, B is one, and in the place where B is zero, A is one. Right? So there is no hierarchy relation between them; neither one is more severe than the other. So I cannot write anything; I leave them that way, with no hierarchy relation between them.

When there is such a relation, notice, what options do we have when there is such a relation? If I assume—I always start from here, from the stronger one—and I assume that the stronger one is alpha. Always. This is a rule: I always assume it this way. That is the beginning. Okay? What will this be? Either two alpha, or alpha plus beta. Right? Two possibilities. Meaning, either this needs—after all, these are results, right? And this too. B and A are results, domains in which I obligate the damage, right? So if for B, to impose it, alpha is enough—therefore it is easier to obligate it, right? Both are one. Then this one is harder to obligate. Why? Because it needs either two alpha, or it needs alpha and beta. Alpha alone is not enough to obligate it. These are two possibilities. If I have no other constraint, I will of course choose the solution of two alpha because it is simpler. Okay? But for example if I have a table like this: A, B, and C. A is alpha, here it will be two alpha, and here I will have to put alpha and beta. There is no other solution. I must add a parameter in such a case, right? Why? Because what would you put here? Just in terms of alpha, what would you put here? Three alpha? How three alpha? If it were three alpha, there should also be a relation like this—reverse, like this. But there is no such relation. If you say to me that beta is two alpha of B—beta is not two alpha; beta is independent, it is another parameter. But B is indeed two alpha. B is two alpha, but C is beta. It is something else. Beta and B are two different things; there is no connection between them. Beta and alpha, yes. Okay, which table is this referring to? No, this is not referring to any table; I am only saying that if we receive a diagram like this, I am saying that here, when I have an arrow like this, that is the datum, that is the basic relation. There is an order relation between columns. When there is an order relation between columns, then the column in which it is easier to impose liability always sits at the head of the arrow. If it is alpha, then the column in which it is harder to impose liability must contain something more than alpha. So it can be either two alpha or alpha and beta. Or alpha, beta, gamma too—it doesn’t matter—but something more, beyond alpha, right?

Now if I have only this, this is my whole diagram, then clearly I will choose this solution because it is simpler. Here there is already an additional parameter, and I do not want an additional parameter. I choose the simplest, right? So I choose that. But in a diagram like this, for example, then from here to here I put two alpha, but here I have no choice—I must add another parameter. Look, it is not beta; it is alpha and beta. Fine? Because each of these two must be stronger than A. And between them there is no relation, because if there were a relation between them, maybe I would put three alpha here, if there were also an arrow like this. But there is no such arrow. Why can’t it be here—alpha and two alpha? Right, to put two alpha and also beta. You can put anything that is greater. You cannot put two alpha, because then there would also be a relation between these two. How does this help before that in choosing between the two tables? Wait, wait, wait. Right now I’m just teaching how to build the model. I have not yet analyzed the tables. In a moment we’ll see.

So okay, that is how you build the model. Now—wait here, here—notice. In a fortiori reasoning, let us just translate this into the language of the tables. The table with entry one was this one; the table with entry zero was this one. Now I forgot the tables—this is what I have. So if this is alpha, then this is two alpha, and here it must be beta, right? Because if there were two alpha here, there should be an arrow. There is no arrow. So this must be a different parameter. That expresses the independence of the columns. So now you see that entry one is simpler because one parameter is enough: alpha and two alpha, with two values. Entry zero is less simple because two parameters are needed to explain the table. So from the diagram you can derive everything. Okay, good.

So now—what is—maybe one more thing—what do these two parameters actually mean? The parameter alpha is the parameter from whose standpoint B is more than A; it is the parameter A has, and from its standpoint B is more than A. The parameter beta is the parameter from whose standpoint A is more than B. It is exactly like those two ingredients we spoke about regarding love of jazz and love of fine literature, where there is no connection between them. They are two different things. That is what a diagram like this expresses: there is no connection between those two things. There are two axes here that do not talk to each other. Okay?

Now let us return to a refutation of an a fortiori argument. We now solve a refutation of an a fortiori argument in the same way. So we draw a table A, B, and C. What is the relation? A goes to B, right? B is stronger than A. Agreed? Exactly like a fortiori reasoning with entry one, right? B is stronger than A; it is greater than it in every row. What is the relation between B and C? The same thing, right? What is the relation between A and C? Reversed, right? No relation. So this is the diagram. Not when they are equal—reversed. Because in this row this is more severe than that, and in that row that is more severe than this. So the diagram is this one, right? What happens here? Let’s see. What happens here is basically that B and C coincide—they are the same point. That is how I mark it, right? And A is another point. There is no relation between it and those two. Right? Now I have to decide which of the two tables is simpler. That is what will tell me which is the correct entry.

So we already know how to do this: alpha, two alpha, alpha and beta. You see now why I did that table before, right? When there is a table like this, those are the solutions. Or vice versa—of course here it could be two alpha and here alpha and beta. It does not matter. Okay? What happens here? This is alpha and this is beta. No, there should be an arrow between two alpha and alpha. There is no arrow. This is alpha and this is beta, right? What does that mean? That they are on the same level of complexity, and therefore this is a refutation. Because the explanation for the two entries is a two-dimensional explanation, an explanation with two parameters, alpha and beta. It is an explanation of the same level of complexity, and therefore one cannot prefer one entry over the other. That is what in plain Hebrew is called a refutation. Right?

Now notice: can we now reverse the a fortiori argument? We said there is a refutation like this, so let us look at an a fortiori argument built like this, not an a fortiori argument built like that. It will not help you at all. Turn around whatever you want. An a fortiori argument is one argument; it is not two arguments. When you look at the level of the microscopic parameters, there is one model that is simpler and one that is less simple, and once there is a refutation, the two models are on the same level of simplicity. There is no dependence on the wording, whether you go by way of the columns or by way of the rows. Here is the first gain from looking through microscopic parameters. We understand very well why you cannot rotate a fortiori arguments. Why in the whole Talmud you will not find the Sages rotating an a fortiori argument. You cannot; it is impossible to rotate an a fortiori argument. What does rotating help? Build the table, build its explanation, and see—it does not work. It will still be a refutation. Rotating an a fortiori argument does not change the table, right? The table itself remains like this. Only how we talk about the table changes.

So what does this actually mean? Notice what this actually means. It means that I am really looking for what this value of B is. That is really what I am looking for. So I say: I do not know on what the value of B depends. Notice: is it alpha—wait—meaning, is it the same value as A’s, or is it not the same value as A’s? In entry one, you see, it is the same value as A’s, right? Only B is alpha; it is stronger than A. Here the a fortiori argument seemingly works because the entry is one. The parameter that matters for obligating in B is the same parameter that matters for obligating in A. Therefore this is an entry that represents the fact that there is an a fortiori argument. But on the other hand, I have an explanation no less good for the situation where there is zero—where there is no a fortiori argument, where it collapses. And here you see: the parameter that matters for B is not at all the parameter that matters for A; it is another parameter. And love of jazz and love of fine literature do not operate with the same component, with the same parameter, with the same ingredient, right? Therefore this is what the refutation does. The refutation says: your severity relation is true, but it is not relevant. Your severity relation is not relevant, because it is in parameter alpha. Parameter alpha exists in B and in A, you are right. B has more alpha than A. But A has beta, and in that respect it is more than B. And now the big question is which of the two parameters, alpha or beta, determines the liability. Regarding B, you do not know.

Okay, since you do not know, that is a refutation. So our assumption is that beta is greater than alpha? No—what? They are independent parameters; there is no greater and smaller between them. This one has alpha and that one has beta, so who said beta is greater than alpha? No one said that—quite the opposite. That is exactly the point: you cannot decide. Therefore this is a refutation. The parameter that matters regarding B and C is not the parameter that matters regarding A. It is a different feature. So the fact that this one has the feature alpha and that one does not have the feature alpha—that is what the a fortiori argument was built on. But on the other hand, this one does not have the feature beta and that one does. And who told you that what matters for your issue is parameter alpha? Maybe it is parameter beta? That is the refutation. Okay.

So this refutation shows us why you cannot rotate an a fortiori argument. Another advantage of the refutation. So let us sum up—maybe one more remark that I have to make here in order to be fair. Sometimes you also need this. There is still some advantage here. To entry zero? What? The parameters are less by a factor of three or two. Both are binary, right? But here I arrive at alpha and two alpha, meaning alpha is a parameter that takes three values—zero, one, and two, right? Here the two parameters are binary; it is either zero or one. Okay, from this it follows, again, that this is why I go back and forth: from this it follows that apparently the number of values of a parameter is not only not important, as we saw before—meaning it does not carry a complexity weight like adding a parameter—here you see that it does not matter at all. Not only is it not important, it is uninteresting. Meaning that if I have two two-parameter explanations, if in one of them the valency is larger, that does not make it less attractive. Why? Intuition? Is there an intuition here about simple explanations? No. At least I do not find one at the moment. This also will not be the final explanation; I will back away from this later. In a moment… But I am trying to build the model with you step by step so that you can see how it is built. In the end it will also be reasonable, I think.

But before we introduced C, then yes, we did relate to the fact that this is alpha and this is two alpha. From that we built the a fortiori argument. No—we related to the fact that here there is alpha and two alpha, and the alternative is alpha and beta. So this is simpler than that. No, I mean before, without the refutation. No, without the refutation there was alpha and beta. There was no C here; there were only B and A, and it was alpha and beta. And I say that alpha and two alpha is a simpler explanation than alpha and beta. Simpler meaning that valency is also a complication, just less of a complication than two parameters. So I am claiming no—valency changes nothing. Not only does it matter less than the addition of a parameter, it does not matter at all. Fine? That is what comes out of here. I will later retract a bit, and therefore… But step by step—I want to go with you one step at a time.